BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Marc Lackenby (University of Oxford) DTSTART:20220615T160000Z DTEND:20220615T170000Z DTSTAMP:20260423T024615Z UID:TQFT/63 DESCRIPTION:Title: Kn ot theory and machine learning\nby Marc Lackenby (University of Oxford ) as part of Topological Quantum Field Theory Club (IST\, Lisbon)\n\nLectu re held in Room 3.10 (3rd floor\, Mathematics Department\, Instituto Super ior Técnico).\n\nAbstract\nKnot theory is divided into several subfields. One of these is hyperbolic knot theory\, which is focused on the hyperbol ic structure that exists on many knot complements. Another branch of knot theory is concerned with invariants that have connections to 4-manifolds\, for example the knot signature and Heegaard Floer homology. In my talk\, I will describe a new relationship between these two fields that was disco vered with the aid of machine learning. Specifically\, we show that the kn ot signature can be estimated surprisingly accurately in terms of hyperbol ic invariants. We introduce a new real-valued invariant called the natural slope of a hyperbolic knot in the 3-sphere\, which is defined in terms of its cusp geometry. Our main result is that twice the knot signature and t he natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This theorem has applicatio ns to Dehn surgery and to 4-ball genus. We will also present a refined ver sion of the inequality where the upper bound is a linear function of the v olume\, and the slope is corrected by terms corresponding to short geodesi cs that have odd linking number with the knot. My talk will outline the pr oofs of these results\, as well as describing the role that machine learni ng played in their discovery.\n\nThis is joint work with Alex Davies\, And ras Juhasz\, and Nenad Tomasev.\n LOCATION:https://researchseminars.org/talk/TQFT/63/ END:VEVENT END:VCALENDAR